I think it's a simple cronjob. I get it done.

Made a bash file with this content:

cd /var/www/html/
sudo chmod 777 -R .
git reset --hard origin/master
git pull
php composer.phar update
cd extensions/Wikibase
git reset --hard origin/master
git pull
php composer.phar update
cd ../WikibaseLexeme/
git reset --hard origin/master
git pull
php composer.phar update

And added it a the crontab that updates it every day. I wait and monitor it for some days to see if it's working as expected and then we can consider this done.

I checked and it works just fine after several days. I added "php maintenance/update.php -q" to handle schema changes too.